A Note on Obata’s Rigidity Theorem

Guoqiang Wu , Rugang Ye

Communications in Mathematics and Statistics ›› 2014, Vol. 2 ›› Issue (3-4) : 231 -252.

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Communications in Mathematics and Statistics ›› 2014, Vol. 2 ›› Issue (3-4) : 231 -252. DOI: 10.1007/s40304-014-0038-6
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A Note on Obata’s Rigidity Theorem

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Abstract

In this note we present various extensions of Obata’s rigidity theorem concerning the Hessian of a function on a Riemannian manifold. They include general rigidity theorems for the generalized Obata equation, and hyperbolic and Euclidean analogs of Obata’s theorem. Besides analyzing the full rigidity case, we also characterize the geometry and topology of the underlying manifold in more general situations.

Keywords

Obata rigidity / Hessian / Sphere theorems / Warped product metric

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Guoqiang Wu, Rugang Ye. A Note on Obata’s Rigidity Theorem. Communications in Mathematics and Statistics, 2014, 2(3-4): 231-252 DOI:10.1007/s40304-014-0038-6

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